Imprecision in Engineering Design y

Imprecision in Engineering DesignErik K.AntonssonEngineering Design Research LaboratoryDivision of Engineering and Applied ScienceCalifornia Institute of TechnologyKevin N.OttoDepartment of Mechanical EngineeringMassachusetts Institute of TechnologyJanuary9,1995AbstractMethods for incorporating imprecision in engineering design decision-making are briefly reviewed and compared.A tutorial is presented on the Method of Imprecision(MoI),a formal method,based on the mathematics of fuzzy sets,for representing andmanipulating imprecision in engineering design.The results of a design cost estima-tion example,utilizing a new informal cost specification,are presented.The MoI canprovide formal information upon which to base decisions during preliminary engineer-ing design and can facilitate set-based concurrent design.IntroductionOne of the most critical problems in engineering design is making early decisions on a sound basis.However,the early stages of design are also the most uncertain,and obtaining preciseis an integral part of the engineering design process. Not imprecision in thought or logic,but rather the intrinsic vagueness of a preliminary, incomplete description.At the concept stage,the design description is nearly completely vague or imprecise(fuzzy).The design process reduces this imprecision until ultimately thefinal description is precise(crisp),except for tolerances,which represent the allowable limits on stochastic manufacturing variations.Despite this evolution of imprecision,engineering design methods and computer aids have nearly all utilized precise information(though some can include stochastic effects).Solid modeling CAD systems,for example,require precise geometry;there is no option to indicate that a dimension is imprecise or only vaguely known.1The need for a methodology to represent and manipulate imprecision is greatest in the early,preliminary phases of engineering design,where the designer is most unsure of the final dimensions and shape,materials and properties,and performance of the completed design.Additionally,the most important decisions,those with the greatest effect on overall cost,are made in these early stages[35,89,103,96].“If a major project is truly innovative,you cannot possibly know its exact costand its exact schedule at the beginning.And if in fact you do know the exactcost and the exact schedule,chances are that the technology is obsolete.”[27]2This paper will review imprecision and uncertainty methods in engineering design,then present a brief tutorial of the Method of Imprecision(MoI),followed by a few recent ad-vances and some thoughts on future research.Review of MethodsMethods to represent uncertain variables as real numbers,and then perform an aggrega-tion(as a sum,product,integral,min,etc.),for decision-making purposes are not new. Uncertainty may be:uncontrolled stochastic variations in variable values,design impreci-sion as described above,variable values to be chosen by optimization,etc.Probability and Bayesian inferencing[37,95,104],Dempster-Shafer theory[83,87],fuzzy sets and trian-gular norms in general[20,42,44,54,56,113],andfinally utility theory[24,26,43]are among the existing formal3methods for representing uncertainty.These methods all represent uncertainty with a range for each variable and a function defined on that range.An illustration is shown in Figure1,where is an uncertain variable and is the uncertainty on variable.They are also similar in that they all conform to thefirst three restrictions of Table1.The axioms shown in Table1have been proposed as the minimum set of restrictions for an engineering design combination(aggregation) function[58],where is the uncertainty associated with the aspect of the design. The discussion below indicates where these theories diverge among themselves and with optimization theory,matrix methods,and the MoI.Imprecision vs.Uncertainty.Uncertainty,which usually represents uncontrolled stochastic variations with the mathematics of probability,is distinct from imprecision.Un-certainty occurs throughout engineering design,in the form of manufacturing variations, material property variations,etc.Including uncertainty in engineering design decision-making can help produce robust designs by assessing the expected size of variations and determining the risk of failure.Many design methods have been developed specifically to address these calculations,including Taguchi’s method,probabilistic optimization,and utility theory.µddFigure1:Example imprecise design variable(boundary conditions)iff(continuity)(idempotency)is used to mean uncertainty in choosing among alternatives.An imprecise variable in preliminary design is a variable that may potentially assume any value within a possible range because the designer does not know,a priori,thefinal value that will emerge from the design process.The nominal value of a length dimension is an example of an imprecise variable.Even though the de-signer is uncertain about what length to specify,she usually has a preference for certain values over others.This preference,which may arise objectively(e.g.,cost or availability of components or materials)or subjectively(e.g.,from experience),is used to quantify the imprecision with which design variables are known.Once one or more alternative design concepts are available,each can be described by a collection of(imprecise)variables.To represent an imprecise variable,,a range of real numbers could be used,in the style of interval analysis[101,102,100].Alternatively,imprecision could be represented by a range as well as a function,,defined on this range to describe the desirability of or preference for particular values,as illustrated in Figure1.In this way variables whose values are not known precisely can be formally represented.Combination Functions.Nearly all formal design methods for representing uncer-tainty or imprecision utilize one or more functions to aggregate information from multiple attributes.The combination calculation performs a trade-off,such that some aspects of a design may contribute more heavily to the combined result than bination functions are also referred to as metrics.A combination function is a formalization of the process of trading-off competing design attributes,and should satisfy the restrictions for engineering design proposed in[58] and shown in bination functions can be divided into two classes:compensat-ing and non-compensating.A compensating combination function(e.g.,sum)will produce an overall measure of a design alternative where aspects that perform well can compensate for aspects that perform poorly.For example,a potential customer of a new car may prefer plenty of legroom and good fuel economy,and be willing to let a bit more legroom partially compensate for poor fuel economy when creating an aggregate evaluation of a particular new car.A non-compensating combination function(e.g.,min)will produce an overall measure of a design alternative that is limited by the most poorly performing aspect.Formalizing the combination of attributes permits trade-off strategies that are deter-mined informally or implicitly to be decided rationally and explicitly.A formal trade-off method also permits design decisions to be clearly understood and recorded for later re-trieval and examination.When a question regarding a particular design trade-off arises at a later stage in the design process,a formal method can provide a clear and complete pic-ture of how the decision was reached.Moreover,the trade-off can be repeated with revised information,thus confirming or refuting the original decision.Methods for representing and manipulating uncertainty and imprecision,and for com-bining multiple attributes of an engineering design,are reviewed below.Utility Theory.Utility theory[24,26,43]was developed to assist in selecting among a choice of distinct actions,given uncertainty(noise)in the outcome of each action.In utility theory,each aspect of a decision(e.g.,each design variable)is assigned a function represent-ing utility,as illustrated in Figure1.The utilities for the individual aspects are aggregated to determine the overall utility,and the combination of variable values that maximizes the overall utility is used.Utility theory is restricted to decision problems in which the individual preferences can be modeled as additive,either with a weighted sum,or with a“multiplicative”form which is also additive but includes nonlinear terms[26].Since the overall metric is additive,utility theory always reflects a compensating strategy,allowing the higher preference of some goals to offset the lower preference of others.Because utility theory is additive,it fails the annihilation restriction shown in Table1. This means that the utility of one aspect of an alternative can be zero,but the overall utility of the alternative will be non-zero.To surmount this difficulty,Thurston has applied utility theory to design by dividing the complete set of goals into two classes:objective constraints and subjective goals[94,92,93].Objective constraints have crisp achievement levels that must be satisfied,and thus become standard constraints as in non-linear programming for-mulations.Subjective goals are those which can be traded-off,and are modeled using stan-dard utility theory.The subjective goals are traded-off among themselves in a compensating manner(using either of the formulations of utility theory),and the objective constraints are traded-off informally(by iteratively refining the constraint values).The overall strategy, therefore,remains informal.Utility theory was originally developed for management decisions,not for engineering design,and requires that all attributes be aggregated into a single goal(utility).Economists generally believe every aspect of a decision can always be translated into a monetary cost. Monetary costs are additive.Aspects which are not additive,or which cannot be“bought off”,are not deemed possible.The axiom of utility theory which creates the demand that a gain in any aspect must be able to compensate for any loss in any other aspect is the Archi-median property.This restriction requires that any decrease in overall preference caused by changes in the performance of one variable must always be able to be balanced by an increase in performance in any of the other variables.Clearly this is not the case in engineer-ing design,as others have argued[11,97].For example,given afixed material,the tensile strength limits cannot be exceeded no matter the reduction in the design’s cost.Material stress simply cannot always be traded-off in a compensating fashion.The Archimedian property and annihilation cannot be simultaneously satisfied.This implies that utility the-ory will not permit a worst case analysis,which is required in many instances in engineering design[33,49].Fuzzy Sets.Fuzzy sets have been used to represent imprecision in(non-design)decision-making[20,42,44,54,56,113].Fuzzy sets are intended to model subjective uncertainty for use in logic,constructing subjectively uncertain versions of“and”and“or”of classical logic.In thefirst paper describing the use of fuzzy sets for decision-making[9],a decision was defined as a convolution of the constraints and goals,using min as a non-compensating metric.They also suggested that at other times simple multiplication of the individual pref-erences might be appropriate.The overall metrics of min and multiplication have been expanded to the more general class of t-norms,first proposed by Menger[51],and reviewed in Dubois and Prade[21]. T-norms are bounded above by min,and are the uncertain version of conjunction.T-norms are not appropriate as an overall design metric because they do not,in general,satisfy the restrictions of Table1.Probabilistic reasoning,Dempster-Shafer theory,and fuzzy sets all employ t-norms[21].Related to each t-norm is an associated t-conorm(or s-norm),which is the uncertain version of disjunction.T-conorms are bounded below by max.The set of functions between t-norms and t-conorms are the mixed connectives,bounded between max and min,and are the class of combination functions used by the MoI and utility theory.Using fuzzy sets for decision-making has received further attention.[22,38,109,110, 111,112],for example,all discuss converting linguistic expressions into fuzzy sets,and then using the fuzzy mathematics to make decisions.An excellent review of fuzzy multi-ple attribute decision-making is presented in[14].Bellman and Zadeh suggested using a weighted sum of the preferences.[39]and[7]also develop additive metrics using fuzzy sets beyond Bellman and Zadeh’s original work.[110]observed the“softness”of multiplication as a connective,and proposed it(and min)in conjunction with weights.Completely independent of fuzzy set formulations,[32](the same year as Zadeh’s initial paper on fuzzy sets)proposed a product of powers as a“desirability function”in chemical process problems.He observed that the annihilation condition is required for engineering design.[15,16]and[69,70,71,72,73]have applied fuzzy set formulations to design optimiza-tion problems in mechanical engineering.[55]has applied fuzzy sets to nuclear radiation cover design.All have used only a non-compensating trade-off strategy for making deci-sions.Other applications of fuzzy optimization are reviewed by[50].Optimization.Other design methodologies exist that do not explicitly represent un-certainty or preferences on variables.For example,optimization formulations(linear,non-linear,integer,and mixed integer programming)[1,4,65,66,74]assume a relationship between preference and the objective function:the lower the function,the higher the pref-erence.Also,a relation is assumed between preference and the constraint functions:if a constraint is satisfied,the preference is high.If any constraint is slightly violated,that con-straint alone dictates the preference for the design is zero.Single objective optimization utilizes a non-compensating strategy:at any point in the space of design variables,one goal determines the preference(either the objective function,or a constraint).Instead of a single objective formulation with constraints,others have proposed multi-objective optimization.Here,strategies are formally explicit only when a norm across the goals is used[79].For example,weighted sum techniques[17,23,25,57,84,88,108] are compensating formulations:the higher performing objectives are averaged with the lower performing objectives,with the incorporation of importance weighting coefficients. As a specific example,the Archimedian“goal programming”[57,80]formulation is a weighted sum technique,with target values and nonlinear weights.Additive metrics have been discussed above in relation to utility theory.Other formulations can be found in[82, 84].[11]and[97],argue that the formulations that fail the annihilation condition(e.g.,addi-tion)are not well suited for engineering design.Typically,multi-objective formulations are used iteratively,without specifying a formal strategy.Such methods have been used in design[5,68].There are also algorithms for such iteration.STEM[10],GDF[28],and the VI algorithm[45],for example,interactively question a decision maker about relative trade-off preferences.Such algorithms are based on an informal overall metric consistent with utility theory[88],and thus exhibit an informal compensating design strategy.Matrix Methods.Concept selection charts[2,3,6,11,67]are commonly used in en-gineering design decision-making.When using a formal chart,alternatives are listed versus evaluation criteria.Each alternative is ranked on each criterion,and the alternative with the best aggregated(weighted sum)score is selected.[67]presents an alternative technique of summing the negative and positive aspects of each alternative,and then making an informal decision based on these ranks.In an analysis of these methods,[52]considers a choice among four alternatives,and demonstrates that four different metrics selected each of the four different alternatives.The choice of decision-making method can entirely change the outcome,which confirms the importance of utilizing a method appropriate for engineering design.The Analytic Hierarchy Process,or AHP,originally developed by Saaty[29,76,77],is a formal method for determining relationships between discrete alternatives,each of which can be rated by one or more attributes.A tutorial example is presented in[31].AHP has an axiomatic foundation[78].Similar to utility theory,this foundation requires that the Archimedian property must be satisfied,which dictates that the annihilation condition is not satisfied.Since the AHP’s ratings are derived from a linear weighted sum,it only implements a compensating trade-off.AHP can only consider discrete alternatives;no continuous variations can be incorpo-rated.Additionally,all alternatives are compared to the lowest performing alternative;AHP includes no ability to indicate that one(or more)alternatives are completely unacceptable, or have violated one or more constraint(s).However,AHP is one of the few methodologies that can incorporate hierarchical objective criteria.Finally,“Quality Function Deployment,”or QFD[2,34]begins by listing the customer requirements for a design on one axis of a chart and the performance metrics for the design on the other axis.Ratings are performed by using a transformation to convert symbols to numerical equivalents,and summing.This summation has the difficulties of the additive metrics discussed above.Probability Methods.In Taguchi’s method[13,36,41,75,86,90,91],the“quality”of a design variable set is defined by the expected variation of a single performance variable from a target value due to uncontrolled stochastic noise.Thus,Taguchi’s methodfindsthe mean of a single performance variable(variation of from),not preference over many design variables and performance variables[59].Experimental design techniques can also be used to determine experimental points in the noise space.See[8,12,40]for a discussion of factorial methods for determining experi-mental points.Alternatively,Monte Carlo simulation[30]could be used for more accuracy.Probabilistic optimization methods can be used to evaluate a solution when noise is present,by evaluating the expected value of the objective function,as presented by[85, Chapter13].The discussion on optimization methods above applies to probabilistic opti-mization methods as well.Necessity Methods.Necessity determines the probability()of a design operating successfully despite uncontrolled variations in one or more parameters over a range(confi-dence interval).This approach can account for the worst case of random disturbances.[98] and[100]have developed a“Labeled Interval Calculus”that consists of interval mathemat-ics with associated“only”or“every”labels.[99]have shown that a set-based approach to design decision-making(using intervals)can facilitate concurrent engineering.Noise can be rated by the expected value,or by the worst case scenario(using necessity). [85,Appendix C]and[81]reject using confidence intervals.Siddall argues that one has difficulty determining the density function and the confidence level.Savage argues against the use of confidence information since he feels one is still obliged to choose the expected value.However,confidence intervals are widely used in practice[33,49],as is modeling to consider the worst case noise[19].Fuzzy Design Methods.Fuzzy design methods combine many of the valuable at-tributes of the methods described above.They are formal(computable)methods for repre-senting and manipulating design imprecision(uncertainty in choosing among alternatives) using the mathematics of fuzzy sets.Several groups are applying fuzzy methods to engi-neering design problems[105,58,53,114,115].Imprecision is represented by a range,and a function defined on this range(),to describe the desirability of or preference for particular values,and to incorporate the de-signer’s experience and judgement into the design evaluation.Non-parametric attributes (such as material choice,color,style,etc.)as well as real-valued attributes(such as physi-cal dimensions,material properties,cost,etc.)can be used.In the Method of Imprecision(MoI)[105,106,63],constraints can be similarly im-precise,permitting the customer to specify preferences over a range of values,rather thana crisp constraint that may be moved by negotiation later in the design process.Because the method was developed specifically for engineering design,the trade-off combination functions meet the restrictions shown in Table1[58].A choice of two basic combina-tion functions is available to aggregate the preferences for the attributes of the design:the (non-compensating)min and a(compensating)product of powers.Because the MoI does not require all attributes to be aggregated into one evaluation metric,evaluations of the various aspects of a design can be made in a hierarchy.For example,safety margin might be traded-off in a non-compensating way among several parts of the design that are subject to loading,and weight and cost might be traded-off in a compensating way.The results of those two trade-offs then might be traded-off with a non-compensating combination function.Because importance weighting can be readily applied, the relative weight of each aspect of the decision can be incorporated into the hierarchy[48].Finally,stochastic uncertainty(such as uncontrolled manufacturing variations)and pos-sibilistic uncertainty and necessity(such as post-manufacturing tuning adjustments)can be incorporated into the design decision-making by utilizing well known expectation calcula-tions[62].Utility theory and the MoI are strongly similar when there is only one goal,and a com-pensating strategy is used[60].When goals are traded-off in a non-compensating manner, and without considering importance weightings,the MoI reduces to a convolution of the constraints and goals used in fuzzy sets for decision-making[9].The Method of ImprecisionThe following sections will present a brief tutorial on how imprecision is used to facilitate decision-making in engineering design using the MoI.Definitions and Notation.Design variables are denoted,and the valid design vari-able values within the design variable space(DVS)form a subset.The set of valid values for is denoted.The preference that a designer has for values of,the th design variable,is represented by a preference function on,called the design preference: .Performance variables are denoted.For each performance variable there must be a mapping such that.The mappings can be any calculation or procedure to measure the performance of a design,including closed-form equations(e.g.,for stress, weight,speed,cost,etc.),iterative solutions,heuristic methods,“black box”calculations, testing of prototypes,or consumer evaluations.The subset of valid performance variable values is mapped from and the set of valid values for is denoted.The perfor-mance variable space(PVS)is the dependent set of performances evaluated for each design in the DVS.In order to compare design alternatives,design preferences are mapped onto the PVS via the extension principle[111],discussed below.Specifications and requirements also embody design imprecision,even though most are written as if they were crisp,e.g.,“This device must have a range of at least250km.”Such a requirement implies that given two designs arbitrarily close together,one with a range of250km and one just below,thefirst would be acceptable but not the second,as shown by the dashed line in Figure2.Specifications and requirements in the real world are commonly fuzzy.Often the designer must ask questions to distinguish the underlying1µp250 kmfuzzycrisprangeFigure 2:Example imprecise functional requirementfuzzy constraint so that the final design will satisfy the customer’s actual requirements even though it may violate the crisp constraint initially given.The fuzziness of constraints and the fuzziness of preliminary design variables are both forms of design imprecision and can be represented in exactly the same way.The customer’s preference (requirements)for values of ,the th performance variable,is represented by a preference function called the functional requirement :.The solid line in Figure 2shows the fuzzy functional requirement.Trade-Off Strategies.The combined preference of the designer and customer for a particular design is represented by an overall preference ,which is a function of the design preferences ,and the functional requirements :(1)Two combination functions ()that satisfy the restrictions of Table 1have been iden-tified:min and a product of powers.Figure 3shows the overall preference obtained by combining a design preference and a functional requirement .Both a compensating trade-off (µp 01µd compensatingnon-compensatingpFigure 3:for compensating and non-compensating trade-offsor established informally by experience,are familiar to engineers in industry[99].Such ac-ceptable intervals correspond to intervals over which preference is greater than zero.Thissuggests that rather than determine the preference at each value of,as shown in Fig-ure1,it may be more natural to determine the intervals in,called-cuts,over whichequals or exceeds certain preference values.The use of intervals encourages the passing of set-based design information betweenengineering groups early in the design process[99],and permits the early release of possi-ble sets of design data from one engineering group to the next in advance of precise designinformation.This approach has many advantages over the traditional“point-by-point”de-sign iteration.The MoI can extend set-based concurrent design by providing preferenceinformation over the possible range of design data.Imprecision Calculations.After specifying design preferences on and func-tional requirements on,and identifying a design trade-off strategy,the next step isto determine the induced values of on(design preferences mapped onto the perfor-mances),given by the extension principle[111]:(2)A simple one-dimensional example of Zadeh’s extension principle is shown in Figure4.The performance achieved for each value of the design variable is given by the function ,which is a curve in this simple example.4The corresponding can be mapped onto,producing:the design preference mapped onto the performance space(as illustrated by the dashed lines in Figure4).For more realistic design problems,each willbe a function of many’s,and each function will be a hyper-surface.An algorithm to compute Zadeh’s extension principle(and thus to calculate)is the Level Interval Algorithm(LIA),first proposed by[18]as the“Fuzzy Weighted Average”algorithm and also called the“Vertex Method”,and extended by[107,64].Once the imprecision on each design variable()is induced onto the PVS,the in-duced preferences are combined with the functional requirements()to obtain an over-all preference().The point(or points)with the highest preference correspond to the performance of the overall most preferred design(s).The design problem is tofind the cor-responding set of design variables()that produce the maximum overall preference ().In the typical engineering design case,where the inverse mapping() doesn’t exist,can still be obtained point by point[46].ExampleAn industrial example of cost estimation for aircraft engine design,utilizing the Engine De-velopment Cost Estimator developed by General Electric Aircraft Engines[47,46],shows how a crisp design cost estimator can be integrated with the MoI.This permits imprecise cost estimates to be developed when only imprecise design data is available.The original example included a formal,and imprecise,functional requirement().The example has recently been extended to include more informal functional requirements.Figure5shows the results of the MoI applied to the engine development cost estimator, using a compensating strategy.Two different design options were simultaneously explored,。